Division Ring Non-Zero Elements form Group

Theorem

If $\left({R, +, \circ}\right)$ is a division ring, then $\left({R^*, \circ}\right)$ is a group.

Proof

• A division ring by definition is a ring with unity, and therefore not null.

• A division ring by definition has no zero divisors, so $\left({R^*, \circ}\right)$ is a semigroup.

• $1_R \in \left({R^*, \circ}\right)$ and so the identity of $\circ$is in $\left({R^*, \circ}\right)$.

• By the definition of a division ring, each element of $\left({R^*, \circ}\right)$ is a unit, and therefore has a unique inverse in $\left({R^*, \circ}\right)$.

Thus $\left({R^*, \circ}\right)$ is a semigroup with an identity and inverses and so is a group.

$\blacksquare$